PomeLemma 20.1.23.476687DD2394 Let $\pi \colon X\rightarrow Z$ be a proper bimeromorphic morphism from $X$ to a Kähler manifold $Z$. Consider non-divisorial closed positive $(1,1)$ currents $T,S$ on $X$ in the same cohomology class. If $T\preceq _{\mathcal{I}} S$, then $\pi _*T\preceq _{\mathcal{I}} \pi _*S$.
Lemma 20.1.23476687DD2394
Proof. By Hironaka's Chow lemma theorem 9.10.8 and 16.8.26, we may assume that $\pi $ is a modification.
By 18.1.38,
\[ \pi ^*\pi _*T=T+F_ T,\quad \pi ^*\pi _*S=S+F_ S, \]
where $F_ T$ and $F_ S$ are effective $\pi $-exceptional divisors. Since $\{ T\} =\{ S\} $, the exceptional divisors $F_ T$ and $F_ S$ have the same cohomology class. Factoring $\pi $ into blow-ups and using proposition 20.1.2 at each step, the classes of $\pi $-exceptional prime divisors are linearly independent in the relative cohomology; hence $F_ T=F_ S$. It follows that
\[ \pi ^*\pi _*T=T+F_ T\preceq _{\mathcal{I}}S+F_ S=\pi ^*\pi _*S. \]
Our assertion then follows from 16.8.26. $\square$
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